Canonical transformation to diagonalize Bosonic Hamiltonian

Question

The Hamiltonian of the system of bosons ($a$, $a^{\dagger}$, $b^{\dagger}$ & $b$ are Bose operators) is: \begin{equation} H=\epsilon_{1} a^{\dagger}a+\epsilon_{2}b^{\dagger}b+\frac{\Delta}{2}\left(a^{\dagger}b^{\dagger}+ba \right) \end{equation}

where $\epsilon_{1}$, $\epsilon_{2}$, and ${\Delta}$ are real and positive, ${\Delta}$ < ($\epsilon_{1}$ + $\epsilon_{2}$). I am trying to find a Canonical Transformation to diagonalize this Hamiltonian. And afterward to find expressions for the eigenenergies and parameters of the transformation. I am not sure whether first I need to switch to any other space like momentum etc and using Bogoliubov Transformation. Any help and hint will be highly appreciated.

Answer 1

You need to make sure the bosonic commutation realtions hold for any basis you choose. For that you need the equivalent of a $z$-Pauli matrix $$\sigma_3 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix},$$ where this is just an example for two bosonic operators, the size of the matrix is 2 x #bosons.

You start by expanding your Hamiltonian to a Nambu space Hamiltonian, i.e. in momentum space this basis would be $ \begin{pmatrix} a_k & b_k & a^\dagger_{-k} & b^\dagger_{-k} \end{pmatrix}$. Now that you have your Hamiltonian written in this basis, you diagonalize the matrix $\sigma_3 H$.

The procedure is specified here.

Answer 2

You could represent $\hat{H}$ as \begin{equation} \hat{H} = \begin{pmatrix}\hat{a}^{\dagger}&\hat{b}^{\dagger}&\hat{b}\end{pmatrix}\underbrace{\begin{pmatrix} \epsilon_{1} & 0&\frac{\Delta}{2}\\ 0&\epsilon_2&0\\ \frac{\Delta}{2}&0&0 \end{pmatrix}}_{\equiv M} \begin{pmatrix}\hat{a}\\\hat{b}\\\hat{b}^{\dagger}\end{pmatrix} \end{equation}

The only thing left to do is to calculate the Eigenvalues and Eigenvectors of $M$